Remark 1.1 The helicity conservation of weak solutions for the compressible Euler equations (REF )
in Onsager's critical space \(B^{\theta }_{p,c(\mathbb {N})}\)  and \(B^{\theta }_{p,\infty }\)  in terms of the velocity and the vorticity is studied in
Theorem REF  and Corollary REF . This is consistent with Moffatt's work [1]}. The regularity condition that \(\text{div\,}v,\text{curl\,}v\in C([0,T];L^{{d+1}}(\Omega ))\)  is necessary to ensure that the helicity is well defined.

